Econometric Analysis - Math Problem Example

www.academia-research.com Sumanta Sanyal d: Introduction This paper is devoted to econometric analysis of data of certain French economic factors for the period 1955-1994. The economic factors are tabulated as annual aggregates and rates in either ordinary or logarithmic forms. They are - 1) Natural logarithm of real per capita consumption (LCF), 2) Natural logarithm of real per capita income (LYF), 3) Real interest rate (RF) and 4) Rate of unemployment (UF). The last two are expressed as proportions. The data has been derived from 'London Metropolitan University's' Website section "Quantitative Research Methods" Module: ECP054C 'Assessment for January, 2007, French Consumption Data (Excel file). Econometric Analysis Econometrics itself means 'economic measurement' (Econometrics, Wikipedia, 2006). Its analysis has two principal purposes - to promote empirical content within particular economic theories and to subject such theories to potentially falsifying tests (Econometrics, Wikipedia, 2006). Taking the first, with the French data as example, it can be said that economic theory holds that consumption should be directly proportion to income while it should be inversely so to interest and unemployment rates. The available data should reflect this and thus provide empirical evidence of the theory's validity. The tests that the data will be subjected to will assist in this quest for empiric authenticity. Such tests should not only provide broad evidence of empiricism but should also assist in determining how much current variance there may be from theoretical reckoning and should thus provide reliable pointers as to why such variance is evident. It is, of course, incumbent upon theorists to interpret why there is variance, if evident after analysis, and this is done at the end of the paper. Methodology To find empiric elements within the variables the first strategy adopted is to assume that a certain relation exists among the variables such that LCF (natural logarithm of real per capita consumption) is taken to be an endogenous variable series that is functionally dependent upon the other variable time series' - LYF (Natural logarithm of real per capita income), RF (real interest rate) and UF (unemployment rate). The following equation expresses a 'Population Linear Function' that is linear both in variables and parameters. The equation is as follows: LCFt = + LYFt + RFt + UFt + ut Here, , , and are the unknown parameters that are assumed to be linear, just as the variables LCF, LYF, RF and UF but this is still an assumption. This, together with the fact that the data has been acquired eccentrically, without due consideration for true series function, does not automatically relate parameters, also called estimators, statistically to their true corresponding values. Therefore, it is necessary to find certain other properties within the variables to enable this. This, however is not evident. Again, thus, some more assumptions have to be made such that, based on these second set of assumptions, certain statistical properties between the estimators and their true corresponding values can be assigned (OLS, Wikipedia, 2006; Estimators and Properties, 2006). This second set of assumptions is as follows. 1. The explanatory set of variables - LYF, RF and UF - is fixed. 2. For all n > 1, >0. Where, x stands for the variable and the mean of the series. This is true for all the variables. 3. =m>0. 4. Zero mean disturbance, E(u)=0. 5. Homoscedasticity: Var() = , is constant for all values of i. 6. Nonautocorrelation, Cov() = 0, where . 7. The error term 'u' has a Gaussian distribution such that, . (Assumptions based on: Estimators and Properties, 2006) A second equation, derived from the same data set and based linearly on the same set of assumptions, is introduced here: LCFt = + LYFt + RFt + UFt + (LCFt - 1) + (LYFt - 1) + (RFt - 1) + (UFt - 1) + ut. Ordinary Least Square (OLS) Method: Both equations are subjected to the OLS method. OLS is a mathematical optimisation technique that strives to bind a set of measured data, as is available here, functionally. It actually minimises the sum of the squares of the ordinate differences (called residuals) from their corresponding points within the data set. The set of assumptions made earlier facilitates application of OLS method on the equations. It has been assumed that the estimators are unbiased and are statistically related to their true corresponding values. This is an essential requirement for application of OSL though it may be, as it is here, that the data series' are not a part of a normal distribution (OLS, Wikipedia, 2006). Tests: A set of tests is run on both the equations next to ascertain the following. 1. Both equations are tested first for signs of first-order autocorrelation with the first-order Breusch-Godfrey Test. 2. Both equations are tested next for non-linear functionality with Ramsey's Reset Test. 3. They are next tested for non-normally distributed residuals with the Jarque-Bera Test. 4. They are tested next for heteroscedasticity with White's Test (taken without cross terms). 5. Lastly, they are tested for first-order ARCH Effects. Results 1st Equation: LCFt = +0.547994410309 LYFt +0.0040565038384637 RFt +0.031560533607779 UFt +0.0023487529695258 + ut Multiple Linear Regression Results Tabulated. Variable Estimators t-Ratios Standard Co-Efficients LYFt 0.547994 86.719944 0.922021 RFt 0.004057 1.266254 0.008006 UFt 0.031561 6.692207 0.079221 Constant 0.002349 7.010642 0 The regression standard error = 0.000407 Critical values (alpha 5%). 1-tail CV at 5% = 1.69 2-tail CV at 5% = 2.03 Misspecification Test Results: Reference Econometrics, Wikipedia, 2006. Extracted on 6th December, 2006, from: http://en.wikipedia.org/wiki/Econometrics Least Squares Method, Wikipedia, 2006. Extracted on 6th December, 2006, from: http://en.wikipedia.org/wiki/Least_squares#History Estimators and Properties, 2006. Extracted on 11th December, 2006, from: http://www.xplore-stat.de/tutorials/xegbohtmlframe29.html London Metropolitan University, French Consumption Data. Extracted on 6th December, 2006, from: http://www.londonmet.ac.uk/depts/efib/people/chris-stewart/quantitative-methods-for-finance-ecp033c.cfm Read More